Polar Function Intersection?

Although spring is around the corner, polar functions crossed my mind last weekend as temperatures fell into the teens. And when mathematicians think of cold, we think of polar functions. Consider the graph shown. How many points of intersection do you see?

Most people would say there is one point of intersection on the vertical axis. But what if I told you the two equations were the following? How many points of intersection are there on the domain of 0 to 2pi?

When you try to solve this system of equations, you get -2 = 1 + sin𝚹 or -3 = sin𝚹. Since the sine ratio can only vary between -1 and 1, this equation has no solution. So why does the graph show a point of intersection? Or does it? Watch the video below and decide for yourself.

I encourage you to play around with Desmos at desmos.com/calculator. Here are three things for you to explore. 

1. What if r = 2 instead of r = -2?
2. Notice that I added the term 0 times sin𝚹  to r = -2. What happens if that is removed?
3. Can you create a pair of polar functions that look like they intersect in 2 or 3 places, but don’t actually intersect.

Share your observations in the comments.